Question: $\int (7 x^6 +7 x^3 +4)\,dx=$ $+C$
Answer: We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int (7 x^6 +7 x^3 +4)\,dx= 7\int x^6\,dx +7\int x^3\,dx +4\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int (7 x^6 +7 x^3 +4)\,dx \\\\ &= 7\int x^6\,dx +7\int x^3\,dx +4\int 1\,dx \\\\ &=7 \dfrac{x^7}{7} +7\dfrac{x^4}{4} +4\dfrac{x^1}{1}+C \\\\ &=x^7 +\dfrac{7}{4} x^4 +4 x+C \end{aligned}$ In conclusion, $\int (7 x^6 +7 x^3 +4)\,dx=x^7 +\dfrac{7}{4} x^4 +4 x+C$